# Electrodynamics from Noncommutative Geometry

March 15, 2011

Within the framework of Connes' noncommutative geometry, the notion of an
almost commutative manifold can be used to describe field theories on compact
Riemannian spin manifolds. The most notable example is the derivation of the
Standard Model of high energy physics from a suitably chosen almost commutative
manifold. In contrast to such a non-abelian gauge theory, it has long been
thought impossible to describe an abelian gauge theory within this framework.
The purpose of this paper is to improve on this point. We provide a simple
example of a commutative spectral triple based on the two-point space, and show
that it yields a U(1) gauge theory. Then, we slightly modify the spectral
triple such that we obtain the full classical theory of electrodynamics on a
curved background manifold.

Keywords:

noncommutative geometry; electrodynamics